Algebra 2 Level 3 Syllabus 2015-2016
... For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasin ...
... For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasin ...
Document
... Proof : It is well known that every computer program may be represented by a bit-string (after all, this is how it’s stored inside). Thus a computer program can be thought of as a bit string. As there are bit-strings yet R is uncountable, there can be no onto function from computer programs to decim ...
... Proof : It is well known that every computer program may be represented by a bit-string (after all, this is how it’s stored inside). Thus a computer program can be thought of as a bit string. As there are bit-strings yet R is uncountable, there can be no onto function from computer programs to decim ...
Set theory and logic
... In Chapter 8 several axiomatic theories which fall within the realm of modern algebra are introduced. The primary purpose is to enable us to give self-contained characterizations in turn of the system of integers, of rational numbers, and, finally, of real numbers. This is clone in the last three se ...
... In Chapter 8 several axiomatic theories which fall within the realm of modern algebra are introduced. The primary purpose is to enable us to give self-contained characterizations in turn of the system of integers, of rational numbers, and, finally, of real numbers. This is clone in the last three se ...
MATHEMATICAL STATEMENTS AND PROOFS In this note we
... more interesting question of how to come up with good and relevant ideas to solve the given problem. (Thus, if we compare mathematics to the game of chess, then in this note we merely aim at teaching the rules of chess, i.e. how the various pieces are allowed to move; we do not try to say very much ...
... more interesting question of how to come up with good and relevant ideas to solve the given problem. (Thus, if we compare mathematics to the game of chess, then in this note we merely aim at teaching the rules of chess, i.e. how the various pieces are allowed to move; we do not try to say very much ...
Document
... This process is called composing the two functions. In general, given two functions u = g(x) and y = f(u), the composition of f with g is y = f(g(x)). We call g the inside function and f the outside function. The process of composing two functions is similar to the process of evaluating a function, ...
... This process is called composing the two functions. In general, given two functions u = g(x) and y = f(u), the composition of f with g is y = f(g(x)). We call g the inside function and f the outside function. The process of composing two functions is similar to the process of evaluating a function, ...
the quadratic functions.
... (Parallel Lines) Recall from Intermediate Algebra that parallel lines have the same slope. (Note that two vertical lines are also parallel to one another even though they have an undefined slope.) Find the line parallel to the given line which passes through the given point. ...
... (Parallel Lines) Recall from Intermediate Algebra that parallel lines have the same slope. (Note that two vertical lines are also parallel to one another even though they have an undefined slope.) Find the line parallel to the given line which passes through the given point. ...
Logical Inference and Mathematical Proof
... Lemma: less important theorem used to prove other theorems. Corollary: theorem that trivially follows another theorem. ...
... Lemma: less important theorem used to prove other theorems. Corollary: theorem that trivially follows another theorem. ...
MA 137 — Calculus 1 for the Life Sciences Exponential and
... Proof: Set y = logb x. By definition, this means that b y = x. Apply now loga (·) to b y = x. We obtain y ...
... Proof: Set y = logb x. By definition, this means that b y = x. Apply now loga (·) to b y = x. We obtain y ...
Lecture Notes on the Lambda Calculus
... 2. Gödel defined the class of general recursive functions as the smallest set of functions containing all the constant functions, the successor function, and closed under certain operations (such as compositions and recursion). He postulated that a function is computable (in the intuitive sense) i ...
... 2. Gödel defined the class of general recursive functions as the smallest set of functions containing all the constant functions, the successor function, and closed under certain operations (such as compositions and recursion). He postulated that a function is computable (in the intuitive sense) i ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.